arXiv subjects

M. V. N. Murthy

Publications and source records attributed to M. V. N. Murthy.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Density of States and Integer Partitions: A Semiclassical Approach

In this review we discuss semi-classical methods that are traditionally used to describe many-body systems in physics, but may also be used to describe partitions of integers in analytic number theory. Specifically, we explore the connection between the methods of statistical mechanics and number partitions. Though the two fields appear very different, their fundamental issues bear a close resemblance. In the former case it is the distribution of a given amount of energy among the particles in an ensemble at a given temperature with well defined properties, while in the latter case it is the way an integer is partitioned into other integers, with or without restrictions. We begin with a discussion of the single-particle quantum density of states, also called the level density, in which we illustrate the connection between the density of states and the classical periodic orbits through the semiclassical trace formula. This is then extended to many particle systems. We show that the asymptotic number partition is reproduced by the average (smooth) part of the level density at discrete integer values of the argument. In the especially interesting case of distinct square partitions, pronounced oscillations are well reproduced by the periodic orbit theory in terms of a few orbits characterised by Pythagorean number triples. We speculate on the connection to Fermat's theorem as to why such regular oscillations (though vanishing asymptotically) exist only in this special case. Finally, we discuss some new results for integer partitions of primes, both unrestricted and distinct.

cond-mat.stat-mech↗

arXiv2026

Comments on Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity technique (arXiv:2507.07180 [hep-ph])

We give our comments on Ref. [1](arXiv:2507.07180) which critiques our idea of exploring quantum statistics to distinguish between Dirac and Majorana neutrinos proposed in some of our earlier works [2-4]. The ad-hoc symmetrization of the Dirac case amplitude square advocated in Eqs. (16) and (35) of [1] has no physical basis and it leads to violation of lepton number in the the standard model for Dirac neutrinos. Therefore, this symmetrization by hand is in principle incorrect.

hep-ph↗

arXiv2024

Comments on "On the Dirac-Majorana neutrinos distinction in four-body decays" (arXiv:2305.14140 [hep-ph], Phys. Rev. D 109, no.3, 033005 (2024))

In arXiv:2305.14140 [hep-ph] the authors analyze the radiative leptonic decay $\ell^- \to ν_\ell \, \overlineν_{\ell'} \, \ell^{\prime -} \, γ$ to distinguish between Dirac and Majorana nature of neutrinos. They utilize the back-to-back kinematics for this purpose, a special kinematic configuration which we first proposed in our paper arXiv:2106.11785 [hep-ph]. Here we point out how and why their analysis of the back-to-back configuration is incorrect. This makes their conclusion and comments invalid and untenable.

hep-ph↗

arXiv2024

Comments on "Can quantum statistics help distinguish Dirac from Majorana neutrinos?" (arXiv:2402.05172 [hep-ph])

In a recent article arXiv:2402.05172 [hep-ph], the authors discuss the question "whether quantum statistics can help distinguish between Dirac and Majorana neutrinos." The paper contains, among other things, an unsubstantiated critique of the results derived in our papers arXiv:2106.11785 [hep-ph] and arXiv:2307.05654 [hep-ph]. One of the criticisms is related to our expression for differential decay rate for the back-to-back neutrino-antineutrino configuration in the decay $B^0 \to μ^- \, μ^+ \, ν_μ\, \overlineν_μ$. We show that the claim is wrong and point out how the correct result was obtained. The second criticism is related to the implementation of the anti-symmetrization as dictated by quantum statistics for Majorana neutrinos and antineutrinos (which are identical, by definition). Any direct observation of the neutrinos, as done in Ref. \cite{Akhmedov:2024}, would project the neutrinos into distinguishable helicity states, thus nullifying all observable effects of quantum statistics. They have missed the point that our procedure holds when the neutrino and antineutrino remain undetected by the detector. In the back-to-back kinematic configuration, one can infer the neutrino energies without directly detecting their identities. This smartly ensures that the quantum statistical effects are not erased. Their overriding assertion that our papers arXiv:2106.11785 [hep-ph] and arXiv:2307.05654 [hep-ph] are incorrect fails to recognize that in both arXiv:2106.11785 [hep-ph] and arXiv:2307.05654 [hep-ph] we also point out generic conditions under which the practical Dirac-Majorana confusion theorem holds. "Clearly there is no confusion over confusion theorem."

hep-ph↗

arXiv2022

Inferring the nature of active neutrinos: Dirac or Majorana?

The nature of a neutrino, whether it is a Dirac type or Majorana type, may be comprehensively probed using their quantum statistical properties. If the neutrino is a Majorana fermion, then by definition it is identical and indistinguishable from the corresponding antineutrino. When a Majorana neutrino and antineutrino are pair produced, the corresponding state has to obey the Pauli principle unlike in the Dirac case. We use this property to distinguish between the two cases using the process $B^0 \to μ^-\,μ^+\,ν_μ\,\barν_μ$. We show that the two cases differ dramatically in a special kinematic scenario where, in the rest frame of the parent $B$ meson, the muons fly away back-to-back (i.e. fly with 3-momenta of equal magnitudes but opposite directions), and so do the neutrino and antineutrino. Unlike any other scenario, we know the energies and magnitudes of $3$-momenta of both the neutrino and the antineutrino in this back-to-back configuration without even directly measuring them. This provides a way of avoiding the constraint imposed by the `practical Dirac-Majorana confusion theorem', as one need not fully integrate over neutrino and antineutrino in this case. As a true signature of the universal principle of quantum statistics which does not depend on the size of the mass of the particle but its spin, the difference between Dirac and Majorana cases in this special kinematic configuration does survive independent of the neutrino mass as long as neutrino mass is nonzero. The analysis presented here is applicable immediately to several other processes with the same final state as in the case of $B^0$ decay without any major change.

hep-ph↗

arXiv2021

On the asymptotic distinct prime partitions of integers

We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple leading-order exponential form given so far in the literature.

math.NT↗

arXiv2018

Semiclassical analysis of distinct square partitions

We study the number $P(n)$ of partitions of an integer $n$ into sums of distinct squares and derive an integral representation of the function $P(n)$. Using semi-classical and quantum statistical methods, we determine its asymptotic average part $P_{as}(n)$, deriving higher-order contributions to the known leading-order expression [M. Tran {\it et al.}, Ann.\ Phys.\ (N.Y.) {\bf 311}, 204 (2004)], which yield a faster convergence to the average values of the exact $P(n)$. From the Fourier spectrum of $P(n)$ we obtain hints that integer-valued frequencies belonging to the smallest Pythagorean triples $(m,p,q)$ of integers with $m^2+p^2=q^2$ play an important role in the oscillations of $P(n)$. Finally we analyze the oscillating part $δP(n)=P(n)-P_{as}(n)$ in the spirit of semi-classical periodic orbit theory [M. Brack and R. K. Bhaduri: {\it Semiclassical Physics} (Bolder, Westview Press, 2003)]. A semi-classical trace formula is derived which accurately reproduces the exact $δP(n)$ for $n > \sim 500$ using 10 pairs of `orbits'. For $n > \sim 4000$ only two pairs of orbits with the frequencies 4 and 5 -- belonging to the lowest Pythagorean triple (3,4,5) -- are relevant and create the prominent beating pattern in the oscillations. For $n > \sim 100,000$ the beat fades away and the oscillations are given by just one pair of orbits with frequency 4.

cond-mat.stat-mech↗

arXiv2017

Physics Potential of the ICAL detector at the India-based Neutrino Observatory (INO)

The upcoming 50 kt magnetized iron calorimeter (ICAL) detector at the India-based Neutrino Observatory (INO) is designed to study the atmospheric neutrinos and antineutrinos separately over a wide range of energies and path lengths. The primary focus of this experiment is to explore the Earth matter effects by observing the energy and zenith angle dependence of the atmospheric neutrinos in the multi-GeV range. This study will be crucial to address some of the outstanding issues in neutrino oscillation physics, including the fundamental issue of neutrino mass hierarchy. In this document, we present the physics potential of the detector as obtained from realistic detector simulations. We describe the simulation framework, the neutrino interactions in the detector, and the expected response of the detector to particles traversing it. The ICAL detector can determine the energy and direction of the muons to a high precision, and in addition, its sensitivity to multi-GeV hadrons increases its physics reach substantially. Its charge identification capability, and hence its ability to distinguish neutrinos from antineutrinos, makes it an efficient detector for determining the neutrino mass hierarchy. In this report, we outline the analyses carried out for the determination of neutrino mass hierarchy and precision measurements of atmospheric neutrino mixing parameters at ICAL, and give the expected physics reach of the detector with 10 years of runtime. We also explore the potential of ICAL for probing new physics scenarios like CPT violation and the presence of magnetic monopoles.

physics.ins-det↗

arXiv2017

On the asymptotic prime partitions of integers

In this paper, we discuss P(n), the number of ways in which a given integer n may be written as a sum of primes. In particular, an asymptotic form P_as(n) valid for n towards infinity is obtained analytically using standard techniques of quantum statistical mechanics. First, the bosonic partition function of primes, or the generating function of unrestricted prime partitions in number theory, is constructed. Next, the density of states is obtained using the saddle-point method for Laplace inversion of the partition function in the limit of large n. This directly gives the asymptotic number of prime partitions P_as(n). The leading term in the asymptotic expression grows exponentially as sqrt[n/ln(n)] and agrees with previous estimates. We calculate the next-to-leading order term in the exponent, porportional to ln[ln(n)]/ln(n), and show that an earlier result in the literature for its coefficient is incorrect. Furthermore, we also calculate the next higher order correction, proportional to 1/ln(n) and given in Eq.(43), which so far has not been available in the literature. Finally, we compare our analytical results with the exact numerical values of P(n) up to n \sim 8 10^6. For the highest values, the remaining error between the exact P(n) and our P_as(n) is only about half of that obtained with the leading-order (LO) approximation. But we also show that, unlike for other types of partitions, the asymptotic limit for the prime partitions is still quite far from being reached even for n \sim 10^7.

math-ph↗

arXiv2015

Stable or Unstable Light Dark Matter

We consider the case of light dark matter ($\sim 10$ GeV). We discuss a simple $Z_2$ model of scalar self-interacting dark matter, as well as a related model of unstable long-lived dark matter which can explain the anomalous Kolar events observed decades ago.

hep-ph↗

arXiv2015

Simulation studies of hadron energy resolution as a function of iron plate thickness at INO-ICAL

We report on a detailed simulation study of the hadron energy resolution as a function of the thickness of the absorber plates for the proposed Iron Calorimeter (ICAL) detector at the India-based Neutrino Observatory (INO). We compare the hadron resolutions obtained with absorber thicknesses in the range 1.5--8 cm for neutrino interactions in the energy range 2--15 GeV, which is relevant to hadron production in atmospheric neutrino interactions. We find that at lower energies, the thickness dependence of energy resolution is steeper than at higher energies, however there is a thickness-independent contribution that dominates at the lower thicknesses discussed in this work. As a result, the gain in hadron energy resolution with decreasing plate thickness is marginal. We present the results in the form of fits to a function with energy-dependent exponent.

physics.ins-det↗

arXiv2013

A phenomenological approach to the equation of state of a unitary Fermi gas

We propose a phenomenological approach for the equation of state of a unitary Fermi gas. The universal equation of state is parametrised in terms of Fermi-Dirac integrals. This reproduces the experimental data over the accessible range of fugacity and normalised temperature, but cannot describe the superfluid phase transition found in the MIT experiment \cite{ku}. The most sensitive data for compressibility and specific heat at phase transition can, however, befitted by introducing into the grand partition function a pair of complex conjugate zeros lying in the complex fugacity plane slightly off the real axis.

cond-mat.stat-mech↗

arXiv2013

Hadron energy response of the Iron Calorimeter detector at the India-based Neutrino Observatory

The results of a Monte Carlo simulation study of the hadron energy response for the magnetized Iron CALorimeter detector, ICAL, proposed to be located at the India-based Neutrino Observatory (INO) is presented. Using a GEANT4 modeling of the detector ICAL, interactions of atmospheric neutrinos with target nuclei are simulated. The detector response to hadrons propagating through it is investigated using the hadron hit multiplicity in the active detector elements. The detector response to charged pions of fixed energy is studied first, followed by the average response to the hadrons produced in atmospheric neutrino interactions using events simulated with the NUANCE event generator. The shape of the hit distribution is observed to fit the Vavilov distribution, which reduces to a Gaussian at high energies. In terms of the parameters of this distribution, we present the hadron energy resolution as a function of hadron energy, and the calibration of hadron energy as a function of the hit multiplicity. The energy resolution for hadrons is found to be in the range 85% (for 1GeV) -- 36% (for 15 GeV).

physics.ins-det↗

arXiv2013

On the higher virial coefficients of a unitary Fermi gas

Third and higher order quantum virial coefficients require the solution of the corresponding quantum many-body problem. Nevertheless, in an earlier paper (Phys. Rev. Lett. 108, 260402 (2012)) we proposed that the higher-order cluster integrals of a dilute unitary fermionic gas may be approximated in terms of the two-body cluster, together with an appropriate suppression factor. Although not exact, this ansatz gave a fair agreement up to fugacity z=6 with the experimentally obtained equation of state. The objective of the present note is to give some physical arguments in favor of this ansatz.

cond-mat.quant-gas↗

arXiv2012

A semiclassical analysis of the Efimov energy spectrum in the unitary limit

We demonstrate that the (s-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest ($n=0$) state, apart from a tiny shift of its zeros that remains constant for large $n$.

quant-ph↗

arXiv2012

The universal equation of state of a unitary fermionic gas

It is suggested that for a fermi gas at unitarity, the two-body bond plays a special role. We propose an equation of state using an ansatz relating the interaction part of the $l$-body cluster to its two-body counterpart. This allows a parameter-free comparison with the recently measured equation of state by the ENS group. The agreement between the two over a range of fugacity ($z<5$ for a homogeneous gas, and $z<10$ for the trapped gas) leads us to perform the calculations of more sensitive quantities measured recently by the MIT group.

cond-mat.stat-mech↗

arXiv2011

Group velocity of neutrino waves

We follow up on the analysis of Mecozzi and Bellini (arXiv:1110:1253v1) where they showed, in principle, the possibility of superluminal propagation of neutrinos, as indicated by the recent OPERA result. We refine the analysis by introducing wave packets for the superposition of energy eigenstates and discuss the implications of their results with realistic values for the mixing and mass parameters in a full three neutrino mixing scenario. Our analysis shows the possibility of superluminal propagation of neutrino flavour in a very narrow range of neutrino parameter space. Simultaneously this reduces the number of observable events drastically. Therefore, the OPERA result cannot be explained in this frame-work.

hep-ph↗

arXiv2010

Exact and asymptotic local virial theorems for finite fermionic systems

We investigate the particle and kinetic-energy densities for a system of $N$ fermions confined in a potential $V(\bfr)$. In an earlier paper [J. Phys. A: Math. Gen. {\bf 36}, 1111 (2003)], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point $\bfr$, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these {\it local virial theorems} (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are supported by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers $N$, we show that they practically are surprisingly accurate also for moderate values of $N$.

math-ph↗